The following method discretizes the state space into a finite number of vintages and a finite number of rungs in the skill ladder. The algorithm can be interpreted as introducing a random element to skill accumulation (see Kushner & Dupuis, 1992, on the approximation of continuous-time models by discrete-time Markov chains). The algorithm is not only useful compute an approximation to the equilibrium, but also to form some intuition about the value function, agent’s paths and other objects of the (continuous) model. The death probabilityδis set to zero to simplify
the exposition; of course, all arguments presented here also apply to the caseδ >0. Throughout, we only consider stationary allocations, i.e. variables depend only on (τ, h) but not ont.
First, construct a discrete grid on the rectangle (0≤τ ≤T,0≤h≤1 as follows: Divide the vintages intoSsub-intervals (of equal size ∆τ) and the experience levels into h sub-intervals (of equal size ∆h). The center points of these intervals are denoted by{τi}T
i=1 and{hj}Sj=1.
To approximate skill approximation choices ˙h, we linearly interpolate the value function between adjacent cells. If the grid is such that workers climb less than ∆h
over a time interval of ∆τ in all cells, than linear interpolation is equivalent to the following “stochastic careers”: Set the probabilityp(τi, hj) that the agent moves one box up (from hj in vintageτi to hj+1 in vintageτi+1, that is) such that the
expected slope of his career equals ˙h(τi, hj), but that it does not exceed one:
p(τi, hj) = min ˙ h(τi, hj)∆τ ∆h,1
This means that in order to be able to replicate very steep slopes in this fashion, we need to make the slope ∆h/∆τ become successively greater as kgrows. I will make the following limiting argument: If we have an infinite sequence of discrete approximations as described above, choose the number of grid points asSk =kS0
andHk =k3/2H
0 (the reason for this choice will become clear later). Now, since
the number of grid points for the hierarchy grows faster than the number of grid points for vintages, the maximal possible slope ∆hk/∆τk will grow to infinity, so any slope ˙hwill be covered from somekon, and all points in the upper-left corner of the rectangle will be reached by some mass from somekon. Of course, for each given grid size there still might be some cells in which the bound 1 is reached.
Consider now how the density of workers evolves on the grid:
n(τi+1, hj) =1−p(τi, hj)n(τi, hj) +p(τi, hj−1)n(τi, hj−1).
Now, introduce the (upward-) difference operators ∆hf(τi, hj) = f(τi, hj+1)−
f(τi, hj) and ∆τf(τi, hj) = f(τi+1, hj)−f(τi, hj) for arbitrary functions f(·,·).
Then we can re-write the above as ∆τn(τ, h) =−∆h
h
n(τ, h−1)p(τ, h−1)i=−n(τ, h−1)∆hp(τ, h−1)
−p(τ, h−1)∆hn(τ, h−1)−∆hn(τ, h−1)∆hp(τ, h−1).
Note that the last term on the right-hand side will become small compared to the others when the grid becomes very fine. In the limit, the equation becomes equiv- alent to the mass-transport PDE (6) that describes the behavior of the densityn.
Production in a vintage ˜Y (excluding the TFP terme−γτi) is calculated as
˜
where again the function f is evaluated in the middle of the corresponding box (τi, hj). This expression converges to ˜Y(n(τi,·)) (under mild conditions) for a given functionn(·) asδh→0.
The discrete counterpart for wages is
w(τi, hj) = exp[−γτi]fj Y˜(τi) n(τi, hj)
!1−ρ
. (24)
Note that this gives the wage rate per unit of time. If we want to calculate the counterpart to wage payments over time a worker spends inside a box (τi, hj), of course we have to multiply this wage rate by ∆τ.
The discrete counterpart of the value function is
V(τi, hj) =w(τi, hj)∆τ+e−(β+δ−γ)∆τV(τi
+1, hj)+ (25) = max ˙ h −c 2h˙ 2∆τ+ ˙h∆τ ∆h | {z } =p e−(β+δ−γ)∆τ∆hV(τi +1, hj) .
Since agents only move upward in equilibrium, we take the upward-difference to approximate theh-derive ofV in the spirit of upwind-differencing.
Solving for the optimal policy gives us ˙ h∗ (τi, hj) =e −(β+δ−γ)∆τ c ∆hV(τi+1, hj) ∆h , (26)
which converges to the optimal policy from the agent’s first-order condition in the continuous case. Plugging back in, we obtain the Bellman equation
V(τi, hj) =w(τi, hj)∆τ+e−(β+δ−γ)∆τV(τi
+1, hj) +e −2(β+δ−γ)∆τ1 c ∆hV(τi+1, hj) ∆h !2 ∆τ.
When dividing this equation by ∆τ and taking the limit as ∆τ →0, we obtain the Hamilton-Jacobi-Bellman equation (HJB) (2) for the continuous case.
I solve the system for a given rectangle with lengthT using an algorithm that is inspired by how a real economy might converge to a steady state under adaptive expectations, assuming some inertia in agents’ actions. Given a distribution of agentsnk (wherekindexes the iterations of the algorithm) over the grid, one can calculate the resulting wages from (24). Using the fact that the marginal value of skill is zero when the vintage dies (i.e. ∆hV(k)(τT
+1, hj= 0 for allj), we can back
out the value function recursively going fromτT back to τ1 using (25), which also
yields optimal policies ˙h∗(k) from (26).
As for the promotion efforts ˙h, we now mix some of the optimal policies into the existing ones: ˙h(k+1)=αh˙k+ (1−α) ˙h∗(k). As for the entry decisions, I send more
mass into the starting points with higher value and less mass into those with higher value. Since wages are inversely related to the density, this algorithm drives the
system towards an equilibrium if the tuning parameters are chosen right. Further work is required to prove that this algorithm is indeed a contraction.
To find T∗
, the vintage horizon that is optimal from the planner’s point of view, I varyT and find a density nT by the algorithm above. I then chooseT∗
as the horizonT that maximizes the planner’s criterion described in the beginning of section 3.
The completeMatlab code used in the calibration and more detailed documen- tation are available from the author upon request.
C
Data
This study uses the weakly anonymous IAB Employment Sample (years 1975-2001). Data access was provided via on-site use at the Research Data Center (FDZ) of the German Federal Employment Agency (BA) at the Institute for Employment Research (IAB) and remote data access.31 The data set is a 2% random sample of all Germans covered by the mandatory public unemployment-insurance scheme.32
Every individual holding a job that fell under this scheme for at least several weeks at any point of the period 1975-2001 was at the same 2% risk of being sampled. For every sampled individual, all available employment spells were collected and included in the data set. Available characteristics include pre-tax earnings, gen- der, age and 3-digit occupation code of the person as well as an identifier of the employer’s establishment and a 3-digit industry classification of the establishment. As is common in the literature, I restrict the sample to males who work full time and are between 20 and 61 years old. For consistency reasons, only workers born in former West Germany are considered. The next section provides a more detailed description of the data and the exclusion restrictions.
The data used in this paper are theEmployment Samples provided by theRe-
search Data Center of the German unemployment office 33. They were collected
for administrative purposes by the mandatory unemployment-insurance (UI) sys- tem in Germany. From 1975 until 2001, spell data about the employment situation were collected at least once yearly from all German employees that were subject to contributions to the unemployment-insurance system. Among full-time employ- ees, this excludes only the self-employed andBeamte (public-sector employees with life-time tenure). On the other hand, data are available for all unemployed workers who were paid benefits out of the UI system — the latter are not used in this paper.
31See Drews, Hamann, K¨ohler, Krug, W¨ubbeke & Autorengemeinschaft ’ITM-
Benutzerhandb¨ucher’ (2006) for an excellent documentation of the IAB Employment Sam- ple.
32The sample does not include tenured public-sector employees and the self-employed;
these groups are not overwhelmingly large so that the data set can be seen as representative of the German labor market.
33In German, the data are succinctly called the: (IAB) Besch¨aftigtenstichprobe, pro-
vided by the Forschungsdatenzentrum (FDZ) of the Bundesagentur f¨ur Arbeit (BA) at theInstitut f¨ur Arbeitsmarkt- und Berufsforschung (IAB).
The sampling design for the IABS is as follows: Every German individual who was subject to paying contributions to the UI system at any point between 1975 and 2001 was sampled at a uniform probability of 2%. Once an individual was sampled, all data from the UI system (work spells and unemployment spells) that could be matched to the individual were included in the sample. The data set consists of more than 12 million spells for more than 1 million individuals.
Individual-specific data include a person’s age, sex and a measure of educa- tion34. For each spell, daily earnings, an establishment ID and the worker’s 3-
digit-level occupation classification are available. At the establishment level, some information is available that was obtained from aggregates over the original set of administrative data before the 2%-sample was drawn. These data include 3- digit-level industry classification, number of employees in the establishment in the respective year, and the first and last date between 1975 and 2001 in which the establishment hired a worker subject to UI contributions.
I only consider records of male individuals who are both older than 20 years and younger than 62 years in the beginning of the year under consideration. I only consider spells coming from theBeH, the database for work relationships – all spells stemming fromLeH, the database for unemployment-insurance payments, are automatically excluded from the sample. Also, I consider only full-time employees (stib<8). Furthermore, all spells that are marked as “geringf¨ugige Besch¨aftigung”
(tax-exempt part-time employment) are dropped.
Note that apprentices and interns areincluded in the sample. This is a deliber- ate choice; since these employees constitute arguably a considerable fraction of the labor force that has no job-specific skills yet, it would not be desirable to discard this information in a study on human-capital accumulation.
Also, I only consider individuals whose first employment is with an establish- ment located in former West Germany. This is done in order to ensure compara- bility of the results before and after 1990.35 Furthermore, some quality checks are
performed on the data: Spells of individuals for whom more than one full-time job is declared are discarded. Also, spells with unreasonably low daily earnings are deleted (below 7 Euros in 2000 Euros per day).
Earnings are adjusted for inflation using the consumer price index for West Germany provided by theBundesbank (the German Central Bank).
Stata programs and documentation on how the moments in section 4 were
obtained in detail are available from the author upon request.
34This education measure is only filled for employment contracts where education infor-
mation is necessary to determine UI contributions or benefits, so its information content is limited.
351990 is the year of German re-unification. Note that individuals born in the former
East who have moved West before their first job are included. This should have no major bearing on the results since it is reasonable to assume that these individuals do not differ systematically from West Germans in terms of earnings potential — the education systems in both parts are of similar quality.